We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the un...We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the unit speed. The models, without or with immigration, are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.展开更多
In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) su...In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.展开更多
In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for th...In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.展开更多
This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properti...This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.展开更多
It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued br...It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.展开更多
Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) ...Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.展开更多
Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual ...Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.展开更多
A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equati...A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.展开更多
The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the sp...The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the spectral radius of Jacobi matrix of its generating function.展开更多
Let{X_(n)}_(n≥0) be a p-type(p≥2)supercritical branching process with immigration and mean matrix M.Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right ei...Let{X_(n)}_(n≥0) be a p-type(p≥2)supercritical branching process with immigration and mean matrix M.Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right eigenvectors v and u.Let ρ>1 and Y_(n)=ρ^(-n)[u·X_(n)-ρ^(n+1)-1/ρ-1(u·λ)],where the vector λ denotes the mean immigration rate.In this paper,we will show that Yn is a martingale and converges to an r.v.Y as n→∞.We study the rates of convergence to 0 as n→∞of P_(i)(|l·X_(n+1)/1·X_(n)-l·(X_(n)M)/1·X_(n)|>ε),P_(i)(|l·X_(n)/1·X_(n)-l·υ/1·υ>ε),P(|Y_(n)-Y|>ε) for any ε>0,i=1,…,p,1=(1,…,1)and l∈R^(p),the p-dimensional Euclidean space.It is shown that under certain moment conditions,the first two decay geometrically,while conditionally on the event Y≥α(α>0)supergeometrically.The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.展开更多
In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions assoc...In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.展开更多
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the proce...We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.展开更多
We study the convergence rates of the harmonic moments for supercritical branching processes with immigration Zn, extending the previous results for non-immigration cases in literature. As a by-product, the large devi...We study the convergence rates of the harmonic moments for supercritical branching processes with immigration Zn, extending the previous results for non-immigration cases in literature. As a by-product, the large deviations for Zn＋1/Zn are also studied. We can see that there is a phase transition in converging rates depending on the generating functions of both branching and immigration.展开更多
In this paper, we study the large deviation for a supercritical branching process with immigration controlled by a sequence of non-negative integer-valued independently identical distributed random variables, improvin...In this paper, we study the large deviation for a supercritical branching process with immigration controlled by a sequence of non-negative integer-valued independently identical distributed random variables, improving the previous results for non immigration processes. We rely heavily on the detail description and limit property of the generating function of immigration processes.展开更多
We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in deta...We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.展开更多
We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every var...We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.展开更多
For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive ...For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P(Zn=kn) with kn=o(mn) as n→∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.展开更多
Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(C...Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.展开更多
In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an ...In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process.展开更多
The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asympto...The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asymptotic estimate of the survival probability at generation n.展开更多
基金supported by the National Key R&D Program of China(2020YFA0712901).
文摘We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the unit speed. The models, without or with immigration, are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.
文摘In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.
文摘In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.
基金supported by NNSF of China(6053408070571079)Open Fundation of SKLSE of Wuhan University (2008-07-03)
文摘This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.
文摘It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.
基金partially supported by the National Nature Science Foundation of China(11601286,11501146)。
文摘Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.
文摘Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.
基金supported by the National Key R&D Program of China(2020YFA0712900)the National Natural Science Foundation of China(11531001).
文摘A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.
文摘The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the spectral radius of Jacobi matrix of its generating function.
基金Supported by the National Natural Sciences Foundations of China(Grants Nos.11771452,11571372)Sciences Foundations of Hunan(Grants No.2017JJ2328)。
文摘Let{X_(n)}_(n≥0) be a p-type(p≥2)supercritical branching process with immigration and mean matrix M.Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right eigenvectors v and u.Let ρ>1 and Y_(n)=ρ^(-n)[u·X_(n)-ρ^(n+1)-1/ρ-1(u·λ)],where the vector λ denotes the mean immigration rate.In this paper,we will show that Yn is a martingale and converges to an r.v.Y as n→∞.We study the rates of convergence to 0 as n→∞of P_(i)(|l·X_(n+1)/1·X_(n)-l·(X_(n)M)/1·X_(n)|>ε),P_(i)(|l·X_(n)/1·X_(n)-l·υ/1·υ>ε),P(|Y_(n)-Y|>ε) for any ε>0,i=1,…,p,1=(1,…,1)and l∈R^(p),the p-dimensional Euclidean space.It is shown that under certain moment conditions,the first two decay geometrically,while conditionally on the event Y≥α(α>0)supergeometrically.The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.
文摘In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.
基金the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012)Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)
文摘We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.
基金This work was supported by the National Natural Science Foundation of China （Grant No. 11371061）.
文摘We study the convergence rates of the harmonic moments for supercritical branching processes with immigration Zn, extending the previous results for non-immigration cases in literature. As a by-product, the large deviations for Zn＋1/Zn are also studied. We can see that there is a phase transition in converging rates depending on the generating functions of both branching and immigration.
文摘In this paper, we study the large deviation for a supercritical branching process with immigration controlled by a sequence of non-negative integer-valued independently identical distributed random variables, improving the previous results for non immigration processes. We rely heavily on the detail description and limit property of the generating function of immigration processes.
基金supported by National Natural Science Foundations of China (Grant Nos. 10771216 and 11071259)
文摘We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.
基金Supported by National Natural Science Foundtation of China (Grant No. 10771185)
文摘We consider harmonic moments of branching processes in general random environments. For a sequence of square integrable random variables, we give some conditions such that there is a positive constant c that every variable in this sequence belong to Ac or A1c uniformly.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11871103)the National Key Research and Development Program of China(No.2020YFA0712900)Research Foundation for Youth Scholars of Beijing Technology and Business University(Grant No.PXM2019_014213_000007).
文摘For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P(Zn=kn) with kn=o(mn) as n→∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.
基金supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciencessupported by the Royal Society Newton International Fellowship and the EU-funded Hungarian(Grant No.EFOP-3.6.1-16-2016-00008)。
文摘Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.
基金Supported by MPG and CAS joint doctoral promotion program and the Klaus Tschira Stiftung through the International Max-Planck Research School
文摘In this paper, we give a variational characterization for the growth rate of a multitype population modelled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process.
基金Supported by the National Natural Science Foundation of China(No.11301133 and 11471218)the Natural Science Foundation of Hebei province(No.A2014202236 and A2014202052)
文摘The class of population-size-dependent branching processes in independent identically distributed random environments is investigated. Under the critical case and appropriate moment assumption, we establish an asymptotic estimate of the survival probability at generation n.
